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CHEOPS observations confirm nodal precession in the WASP-33 system

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read CHEOPS photometry confirms WASP-33b's orbit is precessing, and the derived stellar Love number matches theory.

desk verdict A transparent CHEOPS analysis that makes a useful confirmation but overstates it: photometry alone cannot fix the sign of b, so the 'photometric detection' of precession leans on the Doppler-tomography prior. read the letter →

arxiv 2412.08557 v1 pith:TMI2NXVG submitted 2024-12-11 astro-ph.EP astro-ph.SR

A. M. S. Smith , Sz. Csizmadia , V. Van Grootel , M. Lendl , C. M. Persson , G. Olofsson , D. Ehrenreich , M. N. Günther
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A. Heitzmann S. C. C. Barros A. Bonfanti A. Brandeker J. Cabrera O. D. S. Demangeon L. Fossati J.-V. Harre M. J. Hooton S. Hoyer Sz. Kalman S. Salmon S. G. Sousa Gy. M. Szabó T. G. Wilson Y. Alibert R. Alonso J. Asquier T. Bárczy D. Barrado W. Baumjohann W. Benz N. Billot L. Borsato C. Broeg A. Collier Cameron A. C. M. Correia P. E. Cubillos M. B. Davies M. Deleuil A. Deline B.-O. Demory A. Derekas B. Edwards J. A. Egger A. Erikson A. Fortier M. Fridlund D. Gandolfi K. Gazeas M. Gillon M. Güdel J. Hasiba Ch. Helling K. G. Isaak L. L. Kiss J. Korth K. W. F. Lam J. Laskar A. Lecavelier des Etangs D. Magrin P. F. L. Maxted B. Merín C. Mordasini V. Nascimbeni R. Ottensamer I. Pagano E. Pallé G. Peter D. Piazza G. Piotto D. Pollacco D. Queloz R. Ragazzoni N. Rando H. Rauer I. Ribas N. C. Santos G. Scandariato D. Ségransan A. E. Simon M. Stalport S. Sulis S. Udry S. Ulmer-Moll J. Venturini E. Villaver V. Viotto I. Walter N. A. Walton S. Wolf
This is my paper · ORCID
classification astro-ph.EPastro-ph.SR
keywords nodalprecessionWASP-33btransitimpactparametergravitydarkeningstellarLovenumberCHEOPSphotometryhotJupiterpulsations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that CHEOPS transit photometry can independently confirm the nodal precession of the hot Jupiter WASP-33b, and that the precession can be turned into a measurement of the host star's second-order fluid Love number. The four new transit light curves give an impact parameter that falls on the secular precession curve established by earlier Doppler-tomography work, and combining the precession model's quadrupole moment with a new stellar rotation period yields $k_{2,\star}$ values in agreement with theoretical stellar models. This matters because a Love number measured from precession is a direct probe of how mass is distributed inside the star, and because detecting precession photometrically opens the way for future wide-field space photometry to find more such systems. The same fits measure a true obliquity of $111.3^{+0.2}_{-0.7}$ degrees, consistent with previous measurements, and the paper redetermines stellar parameters and updates the ephemeris. The occultation observations, by contrast, do not yield a robust eclipse depth because the star's pulsations dominate the signal.

What carries the argument

The load-bearing mechanism is the nodal precession of WASP-33b: the star's rapid rotation makes it oblate, and the resulting gravitational quadrupole moment $J_2$ torques the tilted planetary orbit, producing a roughly 700-year sinusoidal drift of the transit impact parameter $b$. The paper fits the CHEOPS transits with a gravity-darkened transit model, which is necessary because the oblate, gravity-darkened stellar disc makes the light curve asymmetric; from those fits it extracts $b$ and the stellar inclination $i_\star$. It then converts the precession model's $J_2$ into the second-order fluid Love number $k_{2,\star}$, a dimensionless measure of how centrally concentrated the stellar mass is, through $k_{2,\star} = 3J_2(\Omega_{\rm crit}/\Omega_{\rm rot})^2$, using the rotation period implied by $i_\star$.

What would settle it

A decisive test would be a Doppler-tomography observation of a WASP-33 transit taken at an epoch where the precession model predicts a significantly negative $b$; Doppler tomography measures both the magnitude and the sign of $b$, so if the sign turned out to be positive, the CHEOPS points would not lie on the published precession curve and the photometric confirmation would fail.

Watch

Extended reading notes

Core claim

The paper's central claim is that space-based transit photometry alone confirms the nodal precession of the hot Jupiter WASP-33b. Fitting four CHEOPS transits with a gravity-darkened model gives a transit impact parameter $b$ that follows the secular curve predicted by the published nodal-precession model, whose roughly 700-year cycle carries $b$ from positive to negative values over the observational window. The precession model supplies the stellar gravitational quadrupole moment $J_2 = (1.36^{+0.15}_{-0.12})\times 10^{-4}$; combining this with the rotation period derived from the fitted stellar inclination gives a second-order fluid Love number $k_{2,\star}=0.0099\pm0.0012$ (or $0.0074\pm0.0011$ using an alternative inclination), in agreement with theoretical stellar models. The same fits measure the true stellar obliquity $\psi = 111.3^{+0.2}_{-0.7}$ degrees, consistent with previous Doppler-tomography values. The paper also redetermines the stellar radius and mass, updates the ephemeris, and finds that the occultation depth cannot be robustly extracted because the stellar pulsations dominate the eclipse signal.

Load-bearing premise

The confirmation assumes the sign of the transit impact parameter is negative, taken from the earlier precession model, because transit photometry cannot distinguish a transit across the northern half of the stellar disc from one across the southern half.

Editorial extensions

If this is right

  • Nodal precession in a hot-Jupiter system can be confirmed from photometry alone, without the spectroscopic Doppler-tomography measurements that previously detected it.
  • Long-baseline space photometry of bright stars should be able to detect and track such precession in real time, extending the sample beyond the few systems known today.
  • The measured Love number $k_{2,\star}=0.0099\pm0.0012$ (or $0.0074\pm0.0011$) agrees with theoretical stellar models, supporting the models' internal-structure predictions for this type of star.
  • The updated ephemeris gives transit-time predictions with 1-sigma uncertainty below 90 seconds for the rest of this decade and below 150 seconds through the 2030s.
  • The transit fits with the obliquity fixed to the Doppler-tomography value constrain the stellar inclination to $i_\star = 100.5\pm8.4$ degrees.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: a simultaneous Doppler-tomography and photometry campaign would turn the sign ambiguity of $b$ into a decisive test, since transit photometry alone cannot distinguish northern from southern transits.
  • Beyond the paper: the failed occultation recovery suggests that eclipse photometry of pulsating hot Jupiters needs long, same-band out-of-eclipse baselines or simultaneous spectroscopy; pulsation frequencies borrowed from another instrument at another epoch do not solve the problem.
  • Beyond the paper: if precession-based Love numbers like this one can be measured for several hot Jupiters around rapid rotators, the ensemble would map how stellar internal structure depends on mass and age, a test stellar models have not yet faced at this precision.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript analyzes four CHEOPS transits and four occultations of WASP-33b, a hot Jupiter orbiting a rapidly rotating delta Scuti star. The transit light curves are modeled with a gravity-darkening prescription, yielding measurements of the transit impact parameter and stellar inclination; the main fiducial fit fixes the sky-projected obliquity lambda to the Doppler-tomography value of Watanabe et al. (2022). The authors report that the CHEOPS impact parameters are consistent with the nodal precession model of Watanabe et al. (2022), combine that model's J2 with a newly derived stellar rotation period to obtain the fluid Love number k2 (0.0099 +/- 0.0012 or 0.0074 +/- 0.0011), and find agreement with theoretical stellar models. They also re-derive stellar parameters from spectra and Gaia data, characterize pulsations from TESS, and show through extensive tests that the occultation depth cannot be reliably extracted from the CHEOPS data.

Significance. If the nodal precession confirmation were robust, the paper would demonstrate a purely photometric detection channel for spin-orbit precession, with direct relevance to PLATO. The study is careful in several respects: the pulsation treatment is systematic and cross-checked with multiple frequency lists; the occultation analysis includes injection-recovery tests and honestly concludes that no reliable depth can be measured; the stellar parameters are re-determined with modern data; and the sign degeneracy of the impact parameter is explicitly acknowledged in Section 7.3. However, the central claim of photometrically confirming nodal precession is weakened by the fact that the CHEOPS b values are sign-degenerate, so the plotted agreement with the Watanabe et al. (2022) curve assumes the sign supplied by that model. The paper is a useful consistency check and a source of improved stellar parameters, but the headline claim is stronger than the data support.

major comments (3)
  1. [Section 7.3 and Fig. 8] The transit light-curve fits cannot distinguish +b from -b, as stated in note 7 of Section 7.3, and the CHEOPS points are plotted with negative b because that sign is expected from the Doppler-tomography model of Watanabe et al. (2022). The agreement of the CHEOPS impact parameters with the precession curve is therefore not an independent photometric confirmation of nodal precession; it is a consistency check between |b| and a curve whose sign was supplied by the model. The abstract's 'confirm' and Section 8's claim that precession 'can be detected from photometry alone' should be tempered to reflect this prior dependence, or the analysis should be extended to show that the photometry can discriminate between the two signs when combined with the model.
  2. [Table 5, case (iii), and Section 7.3] The fiducial CHEOPS measurement b = 0.064+0.075-0.058 is consistent with zero and with a broad range of precession phases, and the authors state that they do not refit the precession model to the CHEOPS data. As presented, the photometric measurement adds little quantitative constraint on the precession parameters, and the word 'confirm' overstates the evidential weight. I recommend adding a quantitative comparison, for example a likelihood ratio between the Watanabe et al. (2022) precession model and a constant-b model evaluated with the CHEOPS points, or an explicit statement of the posterior probability that the CHEOPS b values lie on the model curve given the photometric uncertainties.
  3. [Section 7.1 and Eq. 3] The headline obliquity psi = 111.3+0.2-0.7 deg is quoted from the fit in which lambda is fixed to the Watanabe et al. (2022) Doppler-tomography value, so the quoted uncertainty excludes both the uncertainty in that fixed lambda and the systematic choice among the four degenerate photometric solutions shown in Fig. 6. The free gravity-darkening fit (case ii of Table 5) yields lambda = -148+22-17 deg and psi = 139+12-15 deg with substantially larger errors, and the 'correct' cluster in Fig. 6 is evidently identified using the DT value. The precision of the headline obliquity is therefore largely inherited from the DT prior rather than from the CHEOPS photometry; this should be stated explicitly and the error propagation should include the lambda uncertainty.
minor comments (4)
  1. [Table 5, note] The sentence beginning 'Tthe best ephemeris for planning future observations' contains a typo; 'Tthe' should be 'The'.
  2. [Section 3.1] The word 'temperatute' in the first sentence should be 'temperature'.
  3. [Fig. 6 caption] Please define what is meant by the 'correct' solution in the caption, since the identification of this cluster is used to select the photometric solution in the free gravity-darkening fit.
  4. [Section 7.4] The statement that the Claret (2023) values 'must be multiplied by two' would be easier to verify if the definition of k2 used by Claret were given explicitly, for instance in a short equation or by citing the specific formula.

Circularity Check

2 steps flagged · score 4.0 of 10

Headline obliquity and the photometric 'confirmation' of precession are partly prior-driven: lambda is fixed to the Doppler-tomography value, and the sign of b is imported from the same precession model.

  1. fitted input called prediction [Section 5 (transit fit), Table 5 case (iii), and Abstract Results]
    "Since the data are not able to constrain well the value of λ, we also performed fits where the value of λ = Ω⋆− Ωp (with Ωp = 90◦) is fixed to the value determined from Doppler tomography by Watanabe et al. (2022), λ =−111.5◦. ... We measure an orbital obliquity 111.3+0.2−0.7 degrees, which is consistent with previous measurements made via Doppler tomography."

    The abstract's headline obliquity comes from case (iii), in which the sky-projected obliquity lambda is fixed to the Watanabe et al. (2022) Doppler-tomography value, lambda = -111.5 deg. Because the fitted i_star ~ 100.5 deg and i_p ~ 88.9 deg are both close to 90 deg, Eq. (3) gives psi ~ abs(lambda) almost by construction. The quoted agreement with previous Doppler-tomography measurements is therefore dominated by the input prior, not by an independent photometric determination of the spin-orbit angle. The paper's own free-lambda fit yields psi = 139+12-15 deg, which is less consistent with previous values, demonstrating the sensitivity to the prior.

  2. other [Section 7.3, Fig. 8 and footnote 7; Section 8 conclusions]
    "We note that transit light curve fits are not able to distinguish between +b and −b7, but we plot these points with negative b, since that is what is expected at these epochs from the DT results. ... The value of b measured from the CHEOPS light curves is also consistent with the nodal precession model of Watanabe et al. (2022)."

    The CHEOPS transit fits measure only the magnitude |b|; Table 5 gives b = 0.064+0.075-0.058 as a positive quantity for the fiducial case (iii). To place the CHEOPS points on the Watanabe precession curve in Fig. 8, the sign is set to negative because the DT-based model predicts that sign. The plotted consistency between the CHEOPS measurements and the model therefore uses the model itself to choose which branch of the +b/-b degeneracy to plot; a positive sign would put the same photometric measurements on the opposite side of b = 0, where the model curve is not. Thus the photometry alone can confirm |b| but not the sign or phase of the precession.

full rationale

The derivation chain is largely self-contained on the photometric side: stellar parameters come from independent spectroscopy, Gaia parallax, and isochrone fits; the gravity-darkened transit shape is modelled with TLCM; pulsation handling is cross-checked with multiple frequency lists and a wavelet method; and the occultation depth is honestly reported as non-robust. The Love-number calculation combines the external J2 from Watanabe et al. (2022) with a rotation period from v sin i and i_star, so it is an application of the precession model rather than an independent test of it; no load-bearing self-citation or imported uniqueness theorem appears. The two flagged steps do not make the central measurement vanish: |b| and i_star are genuinely extracted from the CHEOPS light curves, and the magnitude of b does lie near the Watanabe curve at the observed epoch. However, the headline obliquity is computed with lambda fixed to the very DT value it is said to agree with, and the plotted sign of b is imported from the model the paper claims to confirm. These are disclosed priors rather than hidden fits, but they mean the 'confirmation' is only partly independent of its inputs. A score of 4 reflects that partial prior-dependence while acknowledging the real independent photometric content.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claims rest on a small number of adopted inputs: a fixed gravity-darkening coefficient, a Doppler-tomography lambda used in the headline obliquity fit, a literature v sin i_star, and the J2 from the precession model being confirmed. These are not fitted by the CHEOPS data and are the main channels through which external assumptions enter. No new physical entities are introduced.

free parameters (4)
  • Gravity-darkening coefficient beta = 0.23 (fixed)
    Fixed to the value used by Dholakia et al. (2022); the paper notes values can vary (Kalman et al. 2022 free fit gives 0.58 +/- 0.20) and full exploration is out of scope. i_star and psi depend on this choice.
  • Sky-projected obliquity lambda = -111.5 degrees (fixed)
    Adopted from Watanabe et al. (2022) Doppler tomography for the case (iii) fit; the headline psi = 111.3+0.2-0.7 is produced with this prior fixed, so consistency with Doppler tomography is partly by construction.
  • Projected rotational velocity v sin i_star = 86.0 +/- 0.5 km/s
    Adopted from Collier Cameron et al. (2010); combined with i_star to compute the rotation period and hence k2.
  • Stellar quadrupole moment J2 = (1.36 +0.15/-0.12) x 10^-4
    Taken from the Watanabe et al. (2022) precession fit and used in Eq. (4) to compute k2. The paper does not fit J2 to the CHEOPS b data.
assumptions (5)
  • domain assumption Pulsation frequencies measured in TESS S18 are the same in the CHEOPS data, with only amplitudes and phases changing.
    Stated in Section 4.2; underpins the pulsation correction applied to the CHEOPS transits and occultations.
  • domain assumption The TLCM gravity-darkening model with beta = 0.23 accurately describes the stellar surface brightness distribution.
    Section 5; the inference of i_star and psi depends on this model assumption.
  • domain assumption The nodal precession model and J2 of Watanabe et al. (2022) are correct, and the CHEOPS b values are compared to it without refitting.
    Section 7.3; the 'confirmation' is a consistency check against this adopted model.
  • domain assumption Theoretical k2 values from Claret (2023) are reliable references for comparison.
    Section 7.4; used as the reference curve for the Love number agreement.
  • domain assumption Stellar evolutionary models (PARSEC, CLES) and the infrared flux method give unbiased stellar parameters.
    Section 3; the stellar mass, radius, and age used in later calculations depend on these models.

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Cite this review

Pith. "Pith review of CHEOPS observations confirm nodal precession in the WASP-33 system." pith.science (2026). https://pith.science/paper/TMI2NXVG

@misc{pith2026241208557,
  author       = {Pith},
  title        = {Pith review of: CHEOPS observations confirm nodal precession in the WASP-33 system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TMI2NXVG}},
  note         = {Machine review of arXiv:2412.08557}
}
abstract

Aims: We aim to observe the transits and occultations of WASP-33b, which orbits a rapidly-rotating $\delta$ Scuti pulsator, with the goal of measuring the orbital obliquity via the gravity-darkening effect, and constraining the geometric albedo via the occultation depth. Methods: We observed four transits and four occultations with CHEOPS, and employ a variety of techniques to remove the effects of the stellar pulsations from the light curves, as well as the usual CHEOPS systematic effects. We also performed a comprehensive analysis of low-resolution spectral and Gaia data to re-determine the stellar properties of WASP-33. Results: We measure an orbital obliquity 111.3 +0.2 -0.7 degrees, which is consistent with previous measurements made via Doppler tomography. We also measure the planetary impact parameter, and confirm that this parameter is undergoing rapid secular evolution as a result of nodal precession of the planetary orbit. This precession allows us to determine the second-order fluid Love number of the star, which we find agrees well with the predictions of theoretical stellar models. We are unable to robustly measure a unique value of the occultation depth, and emphasise the need for long-baseline observations to better measure the pulsation periods.

Figures

Figures reproduced from arXiv: 2412.08557 by the authors.

Figure 1
Figure 1. CHEOPS photometry of WASP-33 covering the transit (left) and occultation (right) of WASP-33 b. In each case, the PIPE light curve (blue circles) is shown with a small vertical offset with respect to the DRP light curve (green triangles). Larger offsets in flux are applied between each transit / occultation for clarity. Dashed vertical lines indicate the beginning and end of each eclipse. The number to the right of e… view at source ↗
Figure 2
Figure 2. Observed Hα line profile (black) compared to synthetic spec￾tra computed for T⋆,eff values of 7000 (blue), 7100 (red), and 7200 K (green). The lower panel shows the residuals to the three fits. 4.2. Pulsation frequencies measured from TESS data An alternative to measuring the pulsation frequencies directly from the CHEOPS data, is to measure the frequencies from the TESS light curves of WASP-33, which have a much lo… view at source ↗
Figure 3
Figure 3. Fit to the transit data. Upper panel: Phase-folded transit light curve (blue points), overplotted with the best-fitting model with fixed Ω⋆, corresponding to case (iii) of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: Fitted occultation depth for two of the synthetic light curves described in Section 6.2. We fitted for the pulsations using two ap￾proaches: eight frequencies per occultation (left) and twenty frequen￾cies common across all occulations (right). Three different frequenc…
Figure 7
Figure 7. Figure 7: Transit timings compared to a linear ephemeris. The archival points collated by Ivshina & Winn (2022) are shown with blue squares, relative to their linear ephemeris. The timing of our CHEOPS transits are indicated with red circles. −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0…
Figure 8
Figure 8. Figure 8: Impact parameter as a function of time. Doppler tomographic measurements (from Watanabe et al. 2022) are shown in dark blue, pre￾viously published photometric measurements are shown in green, with the two at the same epoch taken from Dholakia et al. (2022) (light green…
Figure 9
Figure 9. Figure 9: Second-order fluid Love number (k2,⋆) of WASP-33 as a function of stellar age. Our two calculated values of k2,⋆ are indicated with dotted red and blue lines, and their 1σ confidence intervals with the shaded red and blue areas. Theoretical k2,⋆ values from Claret (202…

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Pith tools

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